As we all know, puzzles jumble when "they turn to dust when you unbandage them". However, you can make very interesting puzzles by looking at the intermediate steps of unbandaging. I've protyped a program that will, like Jaap's Sphere, let you specify a geometry and in addition to that the possible turns. You can then scramble the puzzle as you like and the pieces are split up as needed to accommodate for your wild turning.
How is this done? Simply, each point in space represents a puzzle piece. For each pixel on the screen, the program calculates what point on the sphere corresponds to it and then backtracks from there: every move is invertible so it can calculate what point originally corresponded to that point on the screen and color it accordingly. Of course this method has enormous disadvantages (you need to keep track of the entire scramble and for every frame, calculate the path of 10000's of points through that scramble) but it's easy to implement and very flexible. It is of course also impossible for the program to tell if the puzzle is solved.

I only built this tool tonight and it's nowhere near finished. It's currently in a state where I might have to start over since my code is so messy, but I see some potential here: instead of just rotations on half-spaces, you could use whatever (invertible) functions you'd want and create truly weird puzzles. This, for instance, this is a truly twisted 3x3x3 where the turning speed depends on the x-coordinate so you get a puzzle with infinitely many layers, all geared together so that each one turns slightly faster than the one above it:

Again, my code is messy and the puzzles are hardcoded (no user input, you have to change the source and program any puzzles on your own) but maybe I can build this into something more usable someday.

Video of me demonstrating the thing.

Incredibly cool!

Okay, I did some work on this and essentially rewrote the entire thing to use a lot cleaner and slightly more efficient code. It now shows you the actual cuts (like on Jaap's sphere) and also highlights the turn you are about to make. Most importantly, I've made it easy to specify an axis system, cut depths and turning angles so it would just take adding the right gui to get a real nice tool.
Check out the new video.

You can download a sample of the program here, it requires the .NET framework. It simulates a fully unbandaged 24-cube.